Strong Gravitational Lensing by Wave Dark Matter Halos

One of the main predictions from Einstein's general relativity is the bending of light as it passes close to a massive body. The deflection angle produced by this effect depends on the mass of the deflector, which then acts like a lens. This deflector may be approximated by a point-like mass, like in the case of a star, but for more massive objects like galaxies it is better to represent them as extended masses that are described by their density profiles. For the purposes of this work, we shall consider galactic lenses, and therefore the density profile described in this section will be representing one galaxy acting as a single lens.

The simplest type of lens is a system with a point mass M located close to the line of sight to a luminous source S. Due to the gravitational field of the point mass, a light ray is deflected in its path to the observer; this is described by the lens equation in the thin-lens approximation. The same approximation also holds for a mass distribution, in which case the lens equation is (Mollerach & Roulet 2002)

$$\begin{eqnarray}&&\beta =\theta -\displaystyle \frac{m(\theta )}{\pi {{\rm{\Sigma }}}_{\mathrm{cr}}{D}_{\mathrm{OL}}^{2}\,\theta },\end{eqnarray} \tag{ 2 }$$

which relates the (unobservable) angle between the line of sight and the path from the observer to the actual position of the source, β, and to the apparent position of the source (the image), θ, and the mass distribution that is causing the lensing m(θ). Also, D_OL is the angular distance from the observer to the lens, which we denote by the subindex (OL). Here we have assumed that m(θ) is the projected mass enclosed in a circle of radius ξ ≡ D_OLθ; more explicitly, we can write

$$\begin{eqnarray}&&m(\xi )=2\pi {\int }_{0}^{\xi }d\mathop{\xi }\limits^{\unicode{x002C6}}\,\mathop{\xi }\limits^{\unicode{x002C6}}\,{\rm{\Sigma }}(\mathop{\xi }\limits^{\unicode{x002C6}}).\end{eqnarray} \tag{ 3a }$$

The projected surface mass density Σ(ξ) can be calculated directly from the (spherically symmetric) density profile ρ(r) of the lensing object as

$$\begin{eqnarray}&&{\rm{\Sigma }}(\xi )=2{\int }_{0}^{{z}_{\max }}{dz}\,\rho (z,\xi ),\end{eqnarray} \tag{ 3b }$$

where $z\equiv \sqrt{{r}^{2}-{\xi }^{2}}$ is a coordinate orthogonal to the line of sight, so that 0 ≤ ξ ≤ r. If the lens system has a finite radius r_max, then ${z}_{\max }=\sqrt{{r}_{\max }^{2}-\xi };$ otherwise, we can put ${z}_{\max }\to \infty$ in the integral given by Equation 3(b).

Let us consider the case in which the density profile ρ(r) has a characteristic density ρ_s and a characteristic radius r_s, such that ρ(r) = ρ_s f(r/r_s), where f is the function that accounts for the shape of the profile. We can then write Equation (2) in the dimensionless form

$$\begin{eqnarray}&&{\beta }_{* }({\theta }_{* })={\theta }_{* }-\lambda \displaystyle \frac{{m}_{* }({\theta }_{* })}{{\theta }_{* }},\end{eqnarray} \tag{ 4 }$$

where the different distances are normalized in terms of r_s: ${\beta }_{* }={D}_{\mathrm{OL}}\beta /{r}_{s},{\theta }_{* }={D}_{\mathrm{OL}}\theta /{r}_{s}$, and then ${\xi }_{* }=\xi /{r}_{s}={\theta }_{* }$. The latter equation means that the normalized variables ${\xi }_{* }$ and ${\theta }_{* }$ can be used interchangeably, and then hereafter we will use ${\theta }_{* }$ as our distance variable.⁵ Likewise, the total mass, as given in Equation 3(a), is normalized as

$$\begin{eqnarray}&&{m}_{* }({\theta }_{* })=\displaystyle \frac{m({\theta }_{* })}{{\rho }_{{\rm{s}}}{r}_{{\rm{s}}}^{3}}=2\pi {\int }_{0}^{{\theta }_{* }}d{\mathop{\theta }\limits^{\unicode{x002C6}}}_{* }\,{\mathop{\theta }\limits^{\unicode{x002C6}}}_{* }\,{{\rm{\Sigma }}}_{* }({\mathop{\theta }\limits^{\unicode{x002C6}}}_{* }).\end{eqnarray} \tag{ 5a }$$

The normalized projected surface mass density, from Equation 3(b), is

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{* }({\theta }_{* })=\displaystyle \frac{{\rm{\Sigma }}({\theta }_{* })}{{\rho }_{{\rm{s}}}{r}_{{\rm{s}}}}=2{\int }_{0}^{{z}_{\max \ast }}{dz}\,f(z,{\theta }_{* }),\end{eqnarray} \tag{ 5b }$$

with $z=\sqrt{{r}_{* }^{2}-{\theta }_{* }^{2}}$ and ${r}_{* }=r/{r}_{s}$. The new parameter λ in Equation (4) is then given by

$$\begin{eqnarray}&&\lambda \equiv \displaystyle \frac{{\rho }_{{\rm{s}}}{r}_{{\rm{s}}}}{\pi {{\rm{\Sigma }}}_{\mathrm{cr}}}={10}^{-3}\displaystyle \frac{0.57}{h}\left(\displaystyle \frac{{\rho }_{{\rm{s}}}{r}_{s}}{{M}_{\odot }{\mathrm{pc}}^{-2}}\right)\displaystyle \frac{{d}_{\mathrm{OL}}{d}_{\mathrm{LS}}}{{d}_{\mathrm{OS}}},\end{eqnarray} \tag{ 6 }$$

where we have defined the reduced (dimensionless) angular distances d_A = D_A H₀/c. The angular diameter distance D_A as a function of redshift is computed in the standard way (see, e.g., Hogg 1999), assuming cosmological model parameters as given by the Planck 2015 results (Planck Collaboration et al. 2016). Equation (6) contains information about the lensing properties of any given model, together with that of the different distances involved in the lens system, namely, between the observer and the lens (OL), between the observer and the source (OS), and between the lens and the source (LS).⁶

One particular case of interest is that of perfect alignment between the luminous source and the lens system for which ${\beta }_{* }({\theta }_{\ast E})=0$. This in turn defines an Einstein ring with radius R_E = D_OLθ_E and an associated angular radius ${\theta }_{E}$. In terms of our normalized variables, we see that the normalized angular Einstein radius ${\theta }_{\ast E}$ directly is the ratio of the Einstein radius to the (characteristic) scale radius of the density profile, ${\theta }_{\ast E}={R}_{E}/{r}_{s}$. Moreover, the angular radius ${\theta }_{\ast E}$ must also be a solution of the equation (see Equation (4))

$$\begin{eqnarray}&&\lambda =\displaystyle \frac{{\theta }_{\ast E}^{2}}{{m}_{* }({\theta }_{\ast E})}.\end{eqnarray} \tag{ 7 }$$

Interestingly enough, Equation (7) shows that the lensing properties of a system with a density profile of the form ρ(r) = ρ_s f(r/r_s) are independent of the density and distance scales and are mostly sensitive to the particular shape of the density profile. The physical parameters of the system are then concentrated in the dimensionless parameter λ in Equation (6), and the latter can be calculated from Equation (7) without any prior knowledge of the given physical scales in the system, namely, ${\rho }_{s}$ and r_s, under only the assumption of perfect alignment (see Figure 1 for an example).

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There is a critical value λ_cr that is the smallest value of λ for which an Einstein ring appears, which must correspond to the limit ${\theta }_{\ast E}\to 0$ in Equation (7). As we shall show now, such a critical value can be calculated analytically in the general case. To avoid the divergence at ${\theta }_{\ast E}=0$ (where ${m}_{* }(0)=0$), we make use of the L'Hôpital rule in Equation (7), and from Equation 3(a) we finally obtain

$$\begin{eqnarray}&&{\lambda }_{\mathrm{cr}}^{-1}=\pi {{\rm{\Sigma }}}_{* }(0)=2\pi {\int }_{0}^{{r}_{\max \ast }}d{\mathop{r}\limits^{\unicode{x002C6}}}_{* }\,f({\mathop{r}\limits^{\unicode{x002C6}}}_{* }),\end{eqnarray} \tag{ 8 }$$

where Σ(0) is the central value of the projected surface mass density given by Equation 3(b). Equation (8) is quite a simple formula for the calculation of λ_cr for any given density profile ρ(r).⁷

As said before, Equation (8) suggests that the critical value λ_cr just depends on the particular shape of the given density profile and no information is necessary about its other physical parameters. The values of λ_crit, calculated from Equation (8) for density profiles that are well known in the literature, are shown in Table 1. For these profiles we also show in Figure 1 the Einstein angle ${\theta }_{\ast E}$ as calculated from Equation (7). As expected, the Einstein angle is the smallest for the WaveDM profile (Equation (10)) alone, which also means that it is the one with the weakest lensing signal.

Table 1.  The Intrinsic Value λ_cr, Calculated from Equation (8) for Halos with Different Density Profiles

Name	Density Profile f(r)	λcr	References
NFW	${\left[{(r/{r}_{s})(1+r/{r}_{s})}^{2}\right]}^{-1}$	0	Wright & Brainerd (2000)
Burkert	$\left[(1+r/{r}_{s})(1{+}^{2}r/{r}_{s}^{2})\right]$	2/π2 ≃ 0.203	Park & Ferguson (2003)
SFDM	$\sin (\pi r/{r}_{s})/(\pi r/{r}_{s})$	0.27	González-Morales et al. (2013)
WaveDM	${(1+{r}^{2}/{r}_{s}^{2})}^{-8}$	$\tfrac{2048}{429{\pi }^{2}}\simeq 0.484$	Schive et al. (2014b), Marsh & Pop (2015)

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We should mention here an additional use of Equation (7): to constrain the free parameters of a given density profile. This relates to the fact that any DM halo characterized by a particular density profile needs to satisfy the constraint λ ≥ λ_cr if it is to produce a lensing signal. Using Equations (6) and (8), the latter statement can be rewritten as

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{s}{r}_{s}}{{M}_{\odot }{\mathrm{pc}}^{-2}}\geqslant {10}^{3}\displaystyle \frac{h}{0.57}\displaystyle \frac{{d}_{\mathrm{OS}}}{{d}_{\mathrm{OL}}{d}_{\mathrm{LS}}}{\lambda }_{\mathrm{cr}}.\end{eqnarray} \tag{ 9 }$$

Equation (9) establishes a minimum value for the (structural) surface density ρ_sr_s of any given DM profile in terms of the measured quantities of a lens system. Although the constraint Equation (9) is satisfied automatically by the NFW profile, for which ${\lambda }_{\mathrm{crit}}=0$, this is not the case for the other profiles listed in Table 1.

For the density profile of WaveDM halos we will consider the model described in Schive et al. (2014a, 2014b), which arises from the study of extensive N-body simulations. The profile consists basically of two parts: one part describing a core sustained by the quantum pressure of the boson particles, also known as the soliton profile, and another part that resembles an NFW-like profile in the outer parts of the halo. As argued in Marsh & Pop (2015), the transition at some radius to an NFW profile must be expected from the change of behavior to CDM on scales larger than the natural length of coherence, which should be proportional to the associated Compton length of the boson particles (in full units, the Compton length is L_C = ℏ/ (mc), where ℏ is the (reduced) Planck's constant and c is the speed of light).

The soliton profile is given by

$$\begin{eqnarray}&&{\rho }_{\mathrm{sol}}(r)=\displaystyle \frac{{\rho }_{s}}{{\left(1+{r}^{2}/{r}_{s}^{2}\right)}^{8}},\end{eqnarray} \tag{ 10 }$$

where r_s and ${\rho }_{s}$ are its characteristic radius and central density contrast, respectively. This profile was first studied in detail in Schive et al. (2014b), although here we are following the nomenclature adopted in Marsh & Pop (2015), where it is also shown that the profile fits well the ground-state solution of the so-called SP system of equations (Ruffini & Bonazzola 1969; Guzman & Urena-Lopez 2004). In this respect, the soliton profile is strongly related to the wave properties (via the Schrödinger equation) of the boson particles.

One important property of the profile given in Equation (10) is that it must also obey the intrinsic scaling symmetry of the SP system (Guzman & Urena-Lopez 2004). If $0\lt \mathop{\lambda }\limits^{\unicode{x002C6}}\ll 1$ is a constant parameter, it can be shown that the central density and radius in the soliton profile are given by

$$\begin{eqnarray}&&{\rho }_{s}={\mathop{\lambda }\limits^{\unicode{x002C6}}}^{4}{m}_{a}^{2}\,{m}_{\mathrm{Pl}}^{2}/4\pi ,\quad {r}_{s}={\left(0.23\mathop{\lambda }\limits^{\unicode{x002C6}}{m}_{a}\right)}^{-1}.\end{eqnarray} \tag{ 11 }$$

This equation suggests that the intrinsic, physical quantities of the soliton profile in Equation (10) are related as shown in Equation (1). This relation will be important later when we discuss the constraints on the boson mass m_a.

For the NFW profile at the outskirts of the galaxy halo we adopt the following parameterization:

$$\begin{eqnarray}&&{\rho }_{\mathrm{NFW}}(r)=\displaystyle \frac{{\rho }_{s}\,{\rho }_{\mathrm{NFW}\ast }}{{\alpha }_{\mathrm{NFW}}\,(r/{r}_{s}){\left(1+{\alpha }_{\mathrm{NFW}}r/{r}_{s}\right)}^{2}}.\end{eqnarray} \tag{ 12 }$$

Notice that in writing Equation (12) we are assuming the following implicit definitions for the scale radius and density, respectively, of the NFW profile: r_NFW = r_s/α_NFW and ${\rho }_{\mathrm{NFW}}={\rho }_{s}\,{\rho }_{\mathrm{NFW}\ast }$, where both α_NFW and ${\rho }_{\mathrm{NFW}\ast }$ are dimensionless numbers.

Unfortunately, there is not precise information in Schive et al. (2014a) about the transition in a galaxy halo from the soliton profile of Equation (10) to the NFW profile of Equation (12) in the general case. Hence, for the present work we adopt the convention for a combined profile as suggested in Marsh & Pop (2015),

$$\begin{eqnarray}&&\rho (r)={\rm{\Theta }}({r}_{\epsilon }-r){\rho }_{\mathrm{sol}}(r)+{\rm{\Theta }}(r-{r}_{\epsilon }){\rho }_{\mathrm{NFW}}(r).\end{eqnarray} \tag{ 13 }$$

where ${\rm{\Theta }}({r}_{\epsilon }-r)$ is the Heaviside step function. Here r is the matching radius where the transition between the individual profiles occurs, and which satisfies the condition ρ(r) =  ρ_s. Notice that $0\lt \epsilon \lt 1$ if the transition between the profiles is to occur at the outskirts of the galaxy halo.

In general terms, and under our parameterization, there are six free parameters in the combined profile (Equation (13)): $({\rho }_{s},{r}_{s},{\rho }_{\mathrm{NFW}\ast },\epsilon ,{r}_{\epsilon },{\alpha }_{\mathrm{NFW}})$. We will now derive two new constraints that arise from the continuity of the combined density profile at the matching radius, which will help us to reduce the number of free parameters.

For a continuous density function, we must impose the condition

$$\begin{eqnarray}&&{\rho }_{\mathrm{sol}}({r}_{\epsilon })=\epsilon {\rho }_{s}={\rho }_{\mathrm{NFW}}({r}_{\epsilon }).\end{eqnarray} \tag{ 14 }$$

When Equation (14) is applied to the soliton profile of Equation (10), we obtain

$$\begin{eqnarray}&&{r}_{\epsilon \ast }={r}_{\epsilon }/{r}_{s}={\left({\epsilon }^{-1/8}-1\right)}^{1/2},\end{eqnarray} \tag{ 15a }$$

which basically establishes the interchangeability of the (dimensionless) matching radius ${r}_{\epsilon \ast }$ and . In the case of the NFW profile (Equation (12)), the continuity condition (Equation (14)) establishes that

$$\begin{eqnarray}&&{\epsilon }^{-1}{\rho }_{\mathrm{NFW}\ast }={\alpha }_{\mathrm{NFW}}\,{r}_{\epsilon \ast }{\left(1+{\alpha }_{\mathrm{NFW}}{r}_{\epsilon \ast }\right)}^{2},\end{eqnarray} \tag{ 15b }$$

which, taking into account Equation 15(a), can be written as

$$\begin{eqnarray}&&{\rho }_{\mathrm{NFW}\ast }=\displaystyle \frac{{\alpha }_{\mathrm{NFW}}\,{r}_{\epsilon \ast }{\left(1+{\alpha }_{\mathrm{NFW}}{r}_{\epsilon \ast }\right)}^{2}}{{\left(1+{r}_{\epsilon \ast }^{2}\right)}^{8}}.\end{eqnarray} \tag{ 15c }$$

Equation 15(c) indicates the (normalized) density ${\rho }_{\mathrm{NFW}\ast }$ that is required for a correct matching between the soliton and NFW profiles, for given values of α_NFW and ${r}_{\epsilon \ast }$.

However, one can see that the continuity constraint (Equation 15(c)) actually shows a hidden degeneracy: once the values of ${\alpha }_{\mathrm{NFW}}$ and ${\rho }_{\mathrm{NFW}\ast }$ are fixed, there can be up to two solutions for the matching radius ${r}_{\epsilon \ast }$. This is a direct consequence of the fact that the crossing of the density profiles (Equations (10) and (12)) can occur at most at two different points, as illustrated in the left panel of Figure 2, which shows normalized density profiles for α_NFW = 0.1 and different values of the normalized density ${\rho }_{\mathrm{NFW}\ast }$.

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Figure 2 (left panel) also shows that there exists a maximum value of ${\rho }_{\mathrm{NFW}\ast }$ beyond which the profiles do not cross each other. This fact can be understood in terms of Equation 15(c), which we evaluate for different values of α_NFW in the right panel of Figure 2. Here we can see that, for each α_NFW, there is always a maximum value of ${\rho }_{\mathrm{NFW}\ast }$, as a function of the matching radius ${r}_{\epsilon \ast }$, that corresponds to the case in which the soliton and NFW density profiles barely touch, as seen in the left panel of Figure 2.

To avoid the hidden degeneracy, and to select always a combined profile with an interior soliton shape, we will choose those cases for which ${r}_{\epsilon \ast }\geqslant {r}_{\epsilon \ast ,\max }$, where ${r}_{\epsilon \ast ,\max }$ is the matching radius corresponding to the maximum value of ${\rho }_{\mathrm{NFW}\ast }$. A straightforward calculation from Equation 15(c) shows that ${r}_{\epsilon \ast ,\max }$ is a root of the cubic equation

$$\begin{eqnarray}&&13{\alpha }_{\mathrm{NFW}}{r}_{\epsilon \ast ,\max }^{3}+15{r}_{\epsilon \ast ,\max }^{2}-3{\alpha }_{\mathrm{NFW}}{r}_{\epsilon \ast ,\max }=1.\end{eqnarray} \tag{ 16 }$$

Although there is a general solution to this equation, it can be shown that the limits for small and large values of α_NFW are

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{{\alpha }_{\mathrm{NFW}}\to 0}{r}_{\epsilon \ast ,\max }=(1/\sqrt{15}),\end{eqnarray} \tag{ 17a }$$

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{{\alpha }_{\mathrm{NFW}}\to \infty }{r}_{\epsilon \ast ,\max }=(\sqrt{3/13}).\end{eqnarray} \tag{ 17b }$$

This means that in absolute terms the maximum value of ${\rho }_{\mathrm{NFW}\ast }$ must be located in the range $0.25\lt {r}_{\epsilon \ast ,\max }\lt 0.48$, which is in agreement with the values observed in the right panel of Figure 2. After all this, it is possible to reduce the number of free parameters that describe the combined profile (13) to only four: ρ_s, r_s, r, and α_NFW. By means of these parameters and the constraints discussed above, the other parameters are fully specified.

Notice that our chosen normalization is such that the physical parameters in the NFW profile (Equation (12)) are given in terms of those in the soliton profile (Equation (10)). This means, for instance, that ${\rho }_{\mathrm{NFW}\ast }\gt 1$ (${\rho }_{\mathrm{NFW}\ast }\lt 1$) is equivalent to ρ_NFW > ρ_s (ρ_NFW < ρ_s), whatever the physical value of ρ_s is. Likewise, we find that ${\alpha }_{\mathrm{NFW}}\lt 1$ (α_NFW > 1) corresponds to ${r}_{\mathrm{NFW}}\gt {r}_{s}$ (r_NFW < r_s), even if the physical value of r_s is not known beforehand. The same will apply for the matching radius, since ${r}_{\epsilon \ast }\gt 1$ (${r}_{\epsilon \ast }\lt 1$) means that matching occurs beyond the soliton radius and then ${r}_{\epsilon }\gt {r}_{s}$ (before the soliton radius and then r < r_s).

It must be noticed also that the prescription above for the matching of the density profiles in Equation (13) means that the NFW part is always subjected to the presence of the central soliton. For instance, the NFW profile can be diluted away if the matching radius ${r}_{\epsilon \ast }\to \infty$ (which also means that ${\rho }_{\mathrm{NFW}\ast }\to 0$), and then the density profile becomes the soliton one alone, ρ(r) ≃ ρ_sol(r). On the other hand, if r_s → 0, the central soliton becomes small but much more massive and denser, because of the scaling symmetry shown in Equation (1), so that it dominates the matter contents over that of the NFW profile. The conclusion here is that under our parameterization the density profile (Equation (13)) can become the soliton profile only if ${r}_{\epsilon \ast }\to \infty$, but it is not possible to do the same for the NFW part; in this sense, the complete profile (Equation (13)) should always be seen as that of a central soliton with a subdominant NFW tail.

We want to stress that the complete profile (13) should not be confused with the so-called cored NFW profile that exists already in the literature. The latter is of the form $\rho {(r)\sim (r/{r}_{s})}^{-\beta }/{(1+r/{r}_{s})}^{2}$ with 0 < β < 1, and whose lensing properties have been analyzed in Wyithe et al. (2001). A comparison of the lenses produced by the cored NFW profile and the WaveDM one is beyond the purpose of this work, as our primary intention is to constrain the parameters in Equation (13) and to obtain from them credible bounds on the mass of the boson particles.

As a final note, we emphasize the convenience of the chosen parameterization in terms of the soliton characteristic quantities, as the soliton and NFW parameters must follow well-defined scaling constraints that are intrinsic to the WaveDM. These scaling properties will then be already explicit in the complete profile (13) when making a comparison of the model with lensing data.

To obtain the lensing properties of the combined profile given by Equation (13), we follow the recipe described in Section 2.1. We first need to compute the projected surface mass density (Equation 3(b)). Because of the presence of the step functions in Equation (13), the integral in Equation 3(b) naturally separates as

$$\begin{eqnarray}\begin{array}{l}{{\rm{\Sigma }}}_{* }({\theta }_{* },{r}_{\epsilon \ast },{\alpha }_{\mathrm{NFW}})\,=\,2\left\{\begin{array}{ll}{\displaystyle \int }_{0}^{\sqrt{{r}_{\epsilon \ast }^{2}-{\theta }_{* }^{2}}}\displaystyle \frac{{dz}}{{\left(1+{\mathop{r}\limits^{\unicode{x002C6}}}^{2}\right)}^{8}}+\displaystyle \frac{{r}_{\epsilon \ast }{\left(1+{\alpha }_{\mathrm{NFW}}{r}_{\epsilon \ast }\right)}^{2}}{{\left(1+{r}_{\epsilon \ast }^{2}\right)}^{8}}\underset{\sqrt{{r}_{\epsilon \ast }^{2}-{\theta }_{* }^{2}}}{\overset{\infty }{\displaystyle \int }}\displaystyle \frac{{dz}}{\mathop{r}\limits^{\unicode{x002C6}}{\left(1+{\alpha }_{\mathrm{NFW}}\mathop{r}\limits^{\unicode{x002C6}}\right)}^{2}}, & {\theta }_{* }\lt {r}_{\epsilon \ast },\\ \displaystyle \frac{{r}_{\epsilon \ast }{\left(1+{\alpha }_{\mathrm{NFW}}{r}_{\epsilon \ast }\right)}^{2}}{{\left(1+{r}_{\epsilon \ast }^{2}\right)}^{8}}{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{dz}}{\mathop{r}\limits^{\unicode{x002C6}}{\left(1+{\alpha }_{\mathrm{NFW}}\mathop{r}\limits^{\unicode{x002C6}}\right)}^{2}}, & {\theta }_{* }\geqslant {r}_{\epsilon \ast }\end{array}\right..\end{array}\end{eqnarray} \tag{ 18 }$$

It should be understood that the integrals in Equation (18) are done along the line of sight. Notice also that we are following our convention in Section 2 for normalized quantities, namely, ${{\rm{\Sigma }}}_{* }={\rm{\Sigma }}/({\rho }_{s}{r}_{s}),{\theta }_{* }=\xi /{r}_{s}$ and $z=\sqrt{{r}_{* }^{2}-{\theta }_{* }^{2}}$. The analytical expressions for the integrals in Equation (18) can be found in the Appendix.

Equation (18) shows that the projected surface mass density only depends on the characteristic radii and densities of the combined density profile (Equation (13)). For instance, if we keep r_s fixed, it can be shown that

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{{r}_{\epsilon \ast }\to \infty }{{\rm{\Sigma }}}_{* }({\theta }_{* },{r}_{\epsilon \ast },{\alpha }_{\mathrm{NFW}})=0.658{\left(1+{\theta }_{* }^{2}\right)}^{-15/2},\end{eqnarray} \tag{ 19 }$$

a result that is obtained from the first branch in Equation (18). Notice that Equation (19) is exactly the result for the soliton profile (Equation (10)) alone. Also, as we have mentioned before, it is not possible to recover the standard result of the surface density for the NFW profile by letting r_s → 0, as in this case the matter content is still dominated by the central soliton.

Going back to the complete profile (Equation (13)), we start with the calculation of the critical value ${\lambda }_{\mathrm{crit}}$ from the analytical formula in Equation (8). The (total) projected surface mass density for the special value ${\theta }_{* }=0$ is obtained from the first branch, in Equation (18), as

$$\begin{eqnarray*}&&\begin{array}{l}{{\rm{\Sigma }}}_{* }(0,{r}_{\epsilon \ast },{\alpha }_{\mathrm{NFW}})\\ =\,2\left[\underset{0}{\overset{{r}_{\epsilon \ast }}{\displaystyle \int }}\displaystyle \frac{{dz}}{{\left(1+{z}^{2}\right)}^{8}}+\displaystyle \frac{{r}_{\epsilon \ast }{\left(1+{\alpha }_{\mathrm{NFW}}{r}_{\epsilon \ast }\right)}^{2}}{{\left(1+{r}_{\epsilon \ast }^{2}\right)}^{8}}{\displaystyle \int }_{{r}_{\epsilon \ast }}^{\infty }\displaystyle \frac{{dz}}{z{\left(1+{\alpha }_{\mathrm{NFW}}z\right)}^{2}}\right],\end{array}\end{eqnarray*}$$

which indicates, together with Equation (7), that the critical value λ_cr of the combined profile (Equation (13)) is a function of ${r}_{\epsilon \ast }$ and α_NFW, and its behavior for different combinations of these parameters is shown in the left panel of Figure 3. Notice that we have taken into account the constraint ${r}_{\epsilon \ast }\geqslant {r}_{\epsilon \ast ,\max }$; see Equation (16). Moreover, it can be seen that the lowest value of λ_cr, for any given value of α_NFW, is indeed attained at ${r}_{\epsilon \ast ,\max }$ as indicated by the vertical lines with the corresponding colors. Not surprisingly, the addition of the NFW outer part helps the soliton profile to achieve small values of λ_crit, which in turn eases the accomplishment of the inequality in Equation (9). In particular, Figure 3 shows that λ_crit → 0 as α_NFW → 0, which means that the combined profile (13) will be able to produce a lensing signal for any nontrivial combination of its parameters ρ_s and r_s.

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In the case of the combined profile the total mass M(r) inside a sphere of any given radius r > r is simply given by the integral

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{M(r)}{{10}^{13}{M}_{\odot }}=3{\left(\displaystyle \frac{{m}_{a}}{{10}^{-22}{\rm{eV}}}\right)}^{-2}{\left(\displaystyle \frac{{r}_{s}}{{\rm{pc}}}\right)}^{-1}\\ \times \,\left[\underset{0}{\overset{{r}_{\epsilon \ast }}{\displaystyle \int }}\displaystyle \frac{{dx}\,{x}^{2}}{{\left(1+{x}^{2}\right)}^{8}}+\displaystyle \frac{{r}_{\epsilon \ast }{\left(1+{\alpha }_{\mathrm{NFW}}{r}_{\epsilon \ast }\right)}^{2}}{{\left(1+{r}_{\epsilon \ast }^{2}\right)}^{8}}\underset{{r}_{\epsilon \ast }}{\overset{{r}_{* }}{\displaystyle \int }}\displaystyle \frac{{dx}\,x}{{\left(1+{\alpha }_{\mathrm{NFW}}x\right)}^{2}}\right].\end{array}\end{eqnarray} \tag{ 20 }$$

In the general case the total mass diverges as $r\to \infty$, whereas for the soliton profile only (which requires ${r}_{\epsilon \ast }\to \infty$) we simply obtain that its total mass M_s is (Guzman & Urena-Lopez 2004; Marsh & Pop 2015; Chen et al. 2017)

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{s}}{{10}^{11}{M}_{\odot }}=7.7{\left(\displaystyle \frac{{m}_{a}}{{10}^{-22}{\rm{eV}}}\right)}^{-2}{\left(\displaystyle \frac{{r}_{s}}{{\rm{pc}}}\right)}^{-1}.\end{eqnarray} \tag{ 21 }$$

In general, we expect from Equation (20) the total mass in the combined profile to be larger than the soliton alone, that is, $M(r)\geqslant {M}_{s}$. However, the value of the total mass M will depend on the upper limit of integration ${r}_{* }$, and the largest values for any given ${r}_{* }$ will be obtained for the case where α_NFW → 0, similar to the case of the critical value λ_crit. The aforementioned general behavior of the total mass M as a function of the free parameters ${r}_{\epsilon \ast }$ and α_NFW is shown in the right panel of Figure 3. For the numerical examples we considered the upper limit of integration ${r}_{* }=20$, for which we then see that that the difference between M and M_s can be as large as three orders of magnitude in the case α_NFW = 0. In other words, the total mass in the WaveDM profile is intrinsically attached to that of its soliton, and then the latter should be large enough if we are going to get the right mass scales in galaxies. This is a nontrivial property, as it shows that any nonzero value of the parameter α_NFW could point out the existence of a soliton core with a non-negligible mass contribution to the lens system (see, e.g., Figure 5 below).

As a first case of study, let us consider the soliton core profile without the external NFW part. There are in this case only two free parameters: ${m}_{a22}$ and ${\theta }_{\ast E}$. In Section 3.2 a Bayesian analysis will be carried out, taking into account the results from this section. The projected mass surface density given by Equation 5(a), with the help of Equation (19), has in this case an analytical expression,

$$\begin{eqnarray}&&{m}_{* }({\theta }_{\ast E})=\displaystyle \frac{2}{13{\lambda }_{\mathrm{crit}}}\,\displaystyle \frac{{\left(1+{\theta }_{\ast E}^{2}\right)}^{13/2}-1}{{\left(1+{\theta }_{\ast E}^{2}\right)}^{13/2}},\end{eqnarray} \tag{ 23 }$$

where ${\lambda }_{\mathrm{crit}}\simeq 0.484$ is the critical value calculated from Equation (8); see also Table 1. Notice that ${m}_{* }(0)=0$, whereas its asymptotic limit is ${m}_{* }(\infty )=2/(13{\lambda }_{\mathrm{crit}})$.

To obtain a basic understanding of the solutions that will be found for the physical parameters, we show in the left panel of Figure 4 the expected behavior of the left-hand side of Equation (22) as a function of the Einstein angle ${\theta }_{\ast E}$. We also show, as the series of horizontal lines, the values of the right-hand side of Equation (22) obtained from the observed data for the galaxies listed in Table 2.

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Figure 4 shows that it will always be possible to identify a value of the Einstein angle ${\theta }_{\ast E}$ for which the left-hand and right-hand sides of Equation (22) are in agreement, irrespective of the value of the boson mass—although as the boson mass increases, the agreement occurs at increasingly large values of ${\theta }_{\ast E}$. For the examples shown in Figure 4, a boson mass of order ${m}_{a22}\simeq 0.02$ seems to fit well the SLACS galaxies listed in Table 2—corresponding to an allowed range for the angular Einstein radius of $5\lt {\theta }_{\ast E}\lt 10$. The latter range can also be translated into an allowed range for the soliton radius and suggests that r_s ∼ kpc for the given example galaxies. However, note that it is always possible to find a solution that matches the left-hand and right-hand sides of Equation (22), for any given value of the boson mass m_a, by a suitably large choice of Einstein angle ${\theta }_{\ast E}$, that is, by choosing r_s → 0. We must recall that the latter condition means that the density profile is dominated by a very massive and compact soliton, but this can be in disagreement with other indications about the actual size of the DM halo in the lens galaxies.

To summarize, given that we have only one observable constraint, the most we can do is first to fix the value of the boson mass m_a and from this to obtain constraints on the remaining free parameters that are consistent with that boson mass. Specifically, by adopting a proposed value for the boson mass m_a in Equations (22) and (23), we can obtain for each galaxy the corresponding best-fit value for ${\theta }_{\ast E}$, and from that the best-fit value for r_s.

The results obtained for our selected sample of galaxies are shown in Table 3 and also plotted in the right panel of Figure 4. The latter figure speaks for itself and shows that the data points for all galaxies lie along the line with a constant soliton mass ${M}_{s}\simeq {10}^{11}\,{M}_{\odot }$ (see Equation (21)), and (as required) all lie below the line that represents the inequality, Equation (9), for the galaxy in Table 2 (J0935–0003) with the most extreme value for the ratio of distances on the right-hand side of Equation (22). The different values obtained for the characteristic radius r_s give an enclosed mass that corresponds closely to the values reported in Auger et al. (2009). Nevertheless, these models are found to be considerably too compact when the characteristic radius and corresponding enclosed mass are considered together. For example, galaxy J0008–0004 has a value for ${M}_{\mathrm{Eins}}=3.1\times {10}^{11}\,{M}_{\odot }$ that is comparable to the value of ${M}_{s}=3.4\times {10}^{11}\,{M}_{\odot }$ obtained using the best-fit parameters of the soliton model. Notwithstanding that the soliton model gives an enclosed mass that is adequate and realistic, we think that the characteristic radius is most definitely not so. This is by taking into consideration the results from rotation curves where the effects of DM are expected to be at larger radii than the luminous part of the galaxy, and this contrasts with the values obtained for the soliton alone where the mean effective radius for galaxy J0008–0004 is observed to be ${r}_{e}\approx 9.6\,{\rm{kpc}}$, which is several orders of magnitude larger than the characteristic radius r_s obtained for any of the different boson masses presented in Table 3, including the samples from the other surveys. Therefore, we think that the soliton profile alone is actually not helping to explain the distribution of DM around the selected galaxies in a consistent way.

Table 3.  Soliton Radius, r_s, Obtained from the Fits to Each Galaxy and for Three Different Values of the Boson Mass m_a

	${m}_{a22}=10$	${m}_{a22}=1$	${m}_{a22}=0.1$
Galaxy	${\mathrm{log}}_{10}({r}_{s}/{\rm{pc}})$
SLACS
J0008–0004	$-{1.67}_{-0.06}^{+0.07}$	${0.33}_{-0.06}^{+0.07}$	${2.33}_{-0.06}^{+0.07}$
J0935–0003	$-{1.73}_{-0.06}^{+0.07}$	${0.27}_{-0.06}^{+0.07}$	${2.27}_{-0.06}^{+0.07}$
J0946+1006	$-{1.59}_{-0.06}^{+0.07}$	${0.41}_{-0.06}^{+0.07}$	${2.41}_{-0.06}^{+0.07}$
J1143–0144	$-{1.41}_{-0.06}^{+0.07}$	${0.59}_{-0.06}^{+0.07}$	${2.59}_{-0.06}^{+0.07}$
J1306+0600	$-{1.46}_{-0.06}^{+0.07}$	${0.54}_{-0.06}^{+0.07}$	${2.54}_{-0.06}^{+0.07}$
J1318–0313	$-{1.62}_{-0.06}^{+0.07}$	${0.37}_{-0.06}^{+0.07}$	${2.37}_{-0.06}^{+0.07}$
LSD
CFRS 03.1077	$-{1.58}_{-0.12}^{+0.17}$	${0.42}_{-0.12}^{+0.17}$	${2.42}_{-0.12}^{+0.17}$
HST 1417+5226	$-{1.7}_{-0.12}^{+0.18}$	${0.30}_{-0.12}^{+0.18}$	${2.30}_{-0.12}^{+0.17}$
SL2S
J220329+020518	$-{1.90}_{-0.05}^{+0.06}$	${0.10}_{-0.05}^{+0.06}$	${2.10}_{-0.05}^{+0.06}$

Note. Note that for SLACS samples, all combinations have a total soliton mass mass contained within the Einstein radius of ${M}_{s}\simeq {10}^{11.5}\,{M}_{\odot }$. The LSD and SL2S samples have a total mass of ${M}_{s}\simeq {10}^{11.8}\,{M}_{\odot }$.

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There are two valuable lessons from the above exercise. The first one is that the soliton core profile alone will always be able to fulfill the lensing constraints even without the consideration of the NFW contribution given the Einstein radius as the only measurement to satisfy. This is not surprising, as the lensing equations can be solved even if we consider a point particle with the required total mass (which formally corresponds to the soliton core profile with ${m}_{a}\to \infty$). The second lesson is that even though the soliton profile may be adequate, formally speaking, to explain the lensing properties of the galaxies in Table 2, we will, in any case, have to consider the NFW outskirts in the complete profile (Equation (13)) in order to satisfy other constraints that suggest that the boson mass should be in the range ${m}_{a22}=1-10$ (see Hui et al. 2017).

Taking into account the above experience gained with the soliton profile alone, we will now consider the following procedure for the complete WaveDM profile. Since the total mass inside the Einstein radius is the only constraint provided by the lens systems, we will fix the values of the boson mass m_a and soliton mass M_s. This approach is considered due to the results from the soliton analysis, where the boson mass can satisfy different values for the Einstein radius, and other studies have found that m_a needs to be in a certain range. For this, we take the following values of the boson mass ${m}_{a22}=0.1,1,10$, and for the soliton mass ${\mathrm{log}}_{10}({M}_{s}/{M}_{\odot })=11.5,10.5,9.5,8.5,7.5$, from which we will calculate the values of r_s by means of Equation (21), which allows us to avoid a possible overcompensation of the soliton mass.

We will adopt a uniform prior for the other parameters over the following ranges: α_NFW = [0:10] and ${r}_{\epsilon \star }=[{r}_{\epsilon \star ,\max }:10]$. Here ${r}_{\epsilon \ast ,\max }$ is found from the cubic Equation (16) for a given value of α_NFW, and the extreme values α_NFW = 10 and ${r}_{\epsilon \ast }=10$ are suggested by Figures 3 and 4.

We will obtain the values of ${\theta }_{\ast E}$ by sampling from a Gaussian distribution, using the relation

$$\begin{eqnarray}&&{\theta }_{\ast E}(p)={\theta }_{\ast {Em}}+\sigma \sqrt{2}{\mathrm{erf}}^{-1}(2p-1),\quad p\in (0,1).\end{eqnarray} \tag{ 24 }$$

The value for ${\theta }_{\ast {Em}}={R}_{E}/{r}_{s}$ is the mean of the distribution using the observed value for the Einstein radius, $\sigma =0.05* \chi$ is the error assigned, and p is a random number sampled from a uniform distribution on the interval [0,1]. The inverse error function is approximated as described in Winitzki (2008). In this way, ${\theta }_{\ast E}$ will not enter into the fitting analysis as an extra variable.

Once the soliton mass is fixed, the rest of the mass that is included within the Einstein radius must be completed by the NFW profile. Because this requires a huge contribution, up to three orders of magnitude more, one sensible consideration is to set the total mass of the lens as composed by a simple representation of luminous matter and the selected model of DM. In a first approximation, the mass corresponding to the baryonic matter is simply a constant value modeled as a point particle. This is done from Equation (2), and then the projected mass for the lens is composed of two parts,

$$\begin{eqnarray}&&m^{\prime} (\theta )=m(\theta )+M^{\prime} ,\end{eqnarray} \tag{ 25 }$$

where m(θ) is the mass from the DM component given by the profile in Equation (13), and $M^{\prime} =f* {,}_{\mathrm{Ein}}{M}_{\mathrm{Ein}}$ is the stellar mass contribution as described in Table 2. These values are normalized accordingly, and then the dimensionless total mass ${m}^{{\prime} }$ is

$$\begin{eqnarray}&&{m}_{* }^{{\prime} }({\theta }_{\ast E},{\alpha }_{\mathrm{NFW}},{r}_{\epsilon \ast })={m}_{* }({\theta }_{\ast E},{\alpha }_{\mathrm{NFW}},{r}_{\epsilon * })+{M}_{* }^{{\prime} },\end{eqnarray} \tag{ 26a }$$

where

$$\begin{eqnarray}&&{M}_{* }^{{\prime} }=0.3208{f}_{\ast ,\mathrm{Ein}}\left(\displaystyle \frac{{M}_{\mathrm{Ein}}}{{M}_{s}}\right).\end{eqnarray} \tag{ 26b }$$

Equation 26(a) is combined with Equation (22) to produce a modified observable that uses the soliton mass directly,

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{s}}{{M}_{\odot }}\,{m}_{* }^{{\prime} }({\theta }_{\ast E},{\alpha }_{\mathrm{NFW}},{r}_{\epsilon \ast })=\displaystyle \frac{7.7\times {10}^{8}}{2.4}\displaystyle \frac{{d}_{\mathrm{OS}}}{{d}_{\mathrm{OL}}{d}_{\mathrm{LS}}}\displaystyle \frac{h}{0.57}{\left(\displaystyle \frac{{R}_{E}}{\mathrm{kpc}}\right)}^{2}.\end{eqnarray} \tag{ 27 }$$

Using the samples mentioned in Section 3, we will try to constrain the free parameters that will satisfy Equation (27). As said before, the information available from the data is the Einstein radius, R_E, the lens distances (${d}_{\mathrm{OL}},{d}_{\mathrm{LS}},{d}_{\mathrm{OS}}$), the lens redshift, and the source redshift. This information is used in the Multinest code (Feroz et al. 2009) to carry out a parameter search for each individual galaxy. We carried out the analysis on the nine galaxies of our subsample of the surveys. Nevertheless, they showed a similar behavior for the range of values of ${f}_{* }$.

For brevity only representative results are shown, as in Figure 5, for the individual cases of galaxies J0935–0003, J0008–0004, and J220329+020518; these cases include the contribution of the luminous matter to the total mass of the lens as in Equation (27). For the purposes of clarity, in each figure we indicate the radius r_s and total mass M_s of the soliton profile. We note that the free parameters ${r}_{\epsilon \ast }$ and ${\alpha }_{\mathrm{NFW}}$ appear well constrained if the soliton mass cannot provide the total mass required by the lens system; in the examples shown, this happens if ${M}_{s}\lt {10}^{11.5}\,{M}_{\odot }$. Second, the credible regions for the parameters in Figure 5 are in agreement with the theoretical expectations discussed in Section 2.2: there is a minimum value for ${r}_{\epsilon \ast }$ due to the constraint imposed by Equation (16), and a maximum value of α_NFW appears due to the maximal contribution of the NFW part of the profile to the total mass in the lens; see also the right panel of Figure 3. Likewise, notice that as α_NFW → 0 the value of the matching radius ${r}_{\epsilon \ast }$ is very well constrained, and this is easily understood from Equation (20): it is ${r}_{\epsilon \ast }$ that determines alone the contribution of the NFW part of the profile to the total mass. Indeed, according to our parameterization in Section 2.2, the NFW part of the density profile, under the limit α_NFW → 0, becomes

$$\begin{eqnarray}&&{\rho }_{\mathrm{NFW}}(r)=\displaystyle \frac{{\rho }_{s}\,{r}_{\epsilon \ast }}{(r/{r}_{s}){\left(1+{r}_{\epsilon \ast }^{2}\right)}^{8}}.\end{eqnarray} \tag{ 28 }$$

Apart from the presence of ${r}_{\epsilon \ast }$ (which in this case is bounded from above, ${r}_{\epsilon \ast }\leqslant 1/\sqrt{15}$; see Equation 17(a)), we also see that the behavior 1/r is the only one that survives from the NFW functional form, and then our results indicate that the outermost behavior 1/r³ is left unconstrained.

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Finally, observe that the value ${\mathrm{log}}_{10}({M}_{s}/{M}_{\odot })=7.5$ is excluded because the soliton mass M_s is so small that the NFW part cannot compensate the required mass for the lens. Recall that there is a matching (continuity) condition for the density profile in which the NFW density ρ_NFW is always smaller than ρ_s, and this condition makes the NFW part of the profile unable to account for the total mass of the lens even in the limit α_NFW → 0.

In summary, if the soliton is allowed to provide enough mass to fulfill the matter contribution in the lens, say, ${M}_{s}\sim {10}^{11.5}\,{M}_{\odot }$, the analysis will select large values for ${r}_{\epsilon \ast }$ so that the NFW tail contribution to the total matter is minimal; see Equation (20). In contrast, if the soliton mass is not large enough, ${M}_{s}\lt {10}^{11.5}\,{M}_{\odot }$, it is then possible to find appropriate pairs $({\alpha }_{\mathrm{NFW}},{r}_{\epsilon \ast })$ for the NFW part of the profile to provide the needed mass for the lens. In this respect, the striped credible regions in Figure 5 represent the degeneracy regions in the plane $({\alpha }_{\mathrm{NFW}},{r}_{\epsilon \ast })$ for the same mass contribution of the NFW tail to the lens system. Thus, we can see that distinct credible regions can be found for the NFW parameters if the soliton mass is ${10}^{8.5}\lt {M}_{s}/{M}_{\odot }\lt {10}^{11.5}$, and that the constraints are in agreement with the semianalytic analysis in Section 2.

Another quantity of interest is the resultant density profile of DM in the lens system. Figure 6 shows examples of the density profiles inferred from the posteriors of galaxy J0008–0004 in Figure 5 for a boson mass ${m}_{a22}=1$. The soliton core is clearly seen in all curves, and so too is the transition to the NFW part of the profile. The corresponding matching radius r, in full units, is selected to be at 15.36, 19.34, 96.94, and 969.4 pc for the soliton masses 10^11.5, 10^10.5, 10^9.5,and ${10}^{8.5}{M}_{\odot }$, respectively. Not surprisingly, the largest core corresponds to the configuration with the lowest soliton mass for which the matching radius is close to the lower bound suggested in Equation 17(a).

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We also report in Figure 7 the results obtained for the lens system J0008–0004 and J220329+020518, for larger or smaller values of the boson mass. For the boson mass ${m}_{a22}=10$ we obtain good constraints on the NFW parameters, but the soliton core is very compact in all cases, although a constraint cannot be found if ${M}_{s}={10}^{6.5}\,{M}_{\odot }$. For the boson mass of ${m}_{a22}=0.1$, we can only obtain well-defined constraints on the NFW parameters when the soliton mass is ${M}_{s}={10}^{10.5}\,{M}_{\odot }$, but not for larger or smaller values. Low values of M_s imply values of the soliton radius r_s that are larger than the Einstein radius, and these kinds of cases are unable to satisfy the lensing constraints. Hence, for a mass of ${m}_{a22}=10$, the soliton is much more compact, and it is not by itself adequate to describe a galaxy. But given the fact that the parameters α_NFW and ${r}_{\epsilon \ast }$ are also well constrained, we conclude that the lensing effect must be mostly attributed to the NFW part. This is not surprising, as we had already indicated in Section 2.3 that strong lensing could be achieved if α_NFW ≪ 1. Moreover, a larger boson mass is also in better agreement with recent cosmological constraints (see Iršič et al. 2017) and with estimations based on satellite galaxies of the Milky Way and Andromeda (see Ureña-López et al. 2017).

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In contrast, we can see that the constraints become more diffuse if we consider a smaller boson mass of ${m}_{a22}=0.1$, although there seems to be some preference for the case in which ${M}_{s}={10}^{10.5}$, which also corresponds to a larger soliton radius. This time the resultant configuration would be in agreement with those found in the statistical analysis carried out in González-Morales et al. (2017), which suggests that satellite galaxies put an upper bound on the boson mass that takes the form ${m}_{a22}\lt 0.4$.